Summary: Explicit provability: the intended semantics
for intuitionistic and modal logic \Lambda
Sergei N. Artemov y
September, 1998
Abstract
The intended meaning of intuitionistic logic is given by the Brouwer­Heyting­Kolmogorov
(BHK) semantics which informally defines intuitionistic truth as provability and specifies
the intuitionistic connectives via operations on proofs. The natural problem of formalizing
the BHK semantics and establishing the completeness of propositional intuitionistic logic
Int with respect to this semantics remained open until recently. This question turned
out to be a part of the more general problem of the intended semantics for G¨odel's modal
logic of provability S4 with the atoms ``F is provable'' which was open since 1933. In this
paper we present complete solutions to both of these problems.
We find the logic of explicit provability (LP) with the atoms ``t is a proof of F '' and
establish that every theorem of S4 admits a reading in LP as the statement about explicit
provability. This provides the adequate provability semantics for S4 along the lines of a
suggestion made by G¨odel in 1938. The explicit provability reading of G¨odel's embedding
of Int into S4 gives the desired formalization of the BHK semantics: Int is shown to
be complete with respect to this semantics. In addition, LP has revealed the relationship
between proofs and types, and subsumes the –­calculus, modal –­calculus and combinatory